Suppose that G ifn a graph. A l-factor is a set of edges of G such that every vertex of G meets exactly one of its edges. Suppose that we have a set Y of l-factors of G such that any two l-factors vf Y have an edge in common. We investigate the following questions:
(1) How large may Y be?
(2) When is there neoessarily an edge contained in all the members of Y?
We answer these questions in the case that G is the complete graph on 2n vertices K2,, or the complete bipartite graph K,,,. In the next section we study the first question; the third section is devoted to the second. In rhe final section we show that B*(K,,,,) = K,,, and B'c.K,,,) = K2,,. We end with some unsolved problems. In the remainder of this section we identify the l-factors of the two graphs, stare our results, and recall the definitions of the space of colorings of G. [4].
B(G). (See
Let the vertices of K2,, be 1,2,3 l l -2~. Then a 1 -factor is a collection of n pairs (a,, a2) l l 0 (a2,,_ ,a& such that every value from 1 to 2n occurs e!,actly once. We call such a collection a pairing, and say that it is composed of n pairs. Km,,, has two sets of tz vertices each, which we shall each number from 1 to n.